From Online Dictionary of Crystallography
Revision as of 13:14, 16 November 2017 by BrianMcMahon (Added German and Spanish translations (U. Mueller))
Sous-groupe normal (Fr). Normalteiler (Ge). Sottogruppo normale (It). 正規部分群 (Ja). Subgrupo normal (Sp).
A subgroup H of a group G is normal in G (H [math]\triangleleft[/math] G) if gH = Hg for any g ∈ G. Equivalently, H ⊂ G is normal if and only if gHg−1 = H for any g ∈ G, i.e. if and only if each conjugacy class of G is either entirely inside H or entirely outside H. This is equivalent to saying that H is invariant under all inner automorphisms of G.
The property gH = Hg means that left and rights cosets of H in G coincide. From this one sees that the cosets form a group with the operation g1H * g2H = g1g2H which is called the factor group or quotient group of G by H, denoted by G/H.
In the special case that a subgroup H has only two cosets in G (namely H and gH for some g not contained in H), the subgroup H is always normal in G.
Connection with homomorphisms
If f is a homomorphism from G to another group, then the kernel of f is a normal subgroup of G. Conversely, every normal subgroup H [math]\triangleleft[/math] G arises as the kernel of a homomorphism, namely of the projection homomorphism G → G/H defined by mapping g to its coset gH.